So I have been reading about Reproducible Kernel Hilbert Spaces (RKHS), and I am confused with how people use it in regards to the kernel trick seen in Machine Learning.
For example, in this blog post (Section 4.), they provide an example saying that they define the map $\phi:\mathcal{X} \rightarrow \mathcal{H}$ to be $\phi(x) = K_x$ where $K_x$ is the reproducing kernel of the element $x \in \mathcal{X}$.
And then we have the kernel function:
$$K(x,z) = \langle \phi(x), \phi(z)\rangle_{\mathcal{H}}$$
Ok so far, I understand this as the "feature map" is a map from the set $\mathcal{X}$ to the hilbert space of functions $\mathcal{H}$.
However, in the example they provide, further down in section 4, they define $\phi(x)$ has a a vector in $\mathbb{R}^{10}$. This confuses me, as I thought that $\phi(x)$ should be an element of the hilbert space of functions $\mathcal{H}$ and not a vector.
Is it because, the vector represents a function by some linear combination of functions?